Muon spin rotation and relaxation studies have been performed on a “111” family of iron-based superconductors, NaFe1−xNixAs, using single crystalline samples with Ni concentrations x=0, 0.4, 0.6, 1.0, 1.3, and 1.5%. Static magnetic order was characterized by obtaining the temperature and doping dependences of the local ordered magnetic moment size and the volume fraction of the magnetically ordered regions. For x=0 and 0.4%, a transition to a nearly-homogeneous long range magnetically ordered state is observed, while for x≳0.4% magnetic order becomes more disordered and is completely suppressed for x=1.5%. The magnetic volume fraction continuously decreases with increasing x. Development of superconductivity in the full volume is inferred from Meissner shielding results for x≳0.4%. The combination of magnetic and superconducting volumes implies that a spatially-overlapping coexistence of magnetism and superconductivity spans a large region of the T−x phase diagram for NaFe1−xNixAs. A strong reduction of both the ordered moment size and the volume fraction is observed below the superconducting TC for x=0.6, 1.0, and 1.3%, in contrast to other iron pnictides in which one of these two parameters exhibits a reduction below TC, but not both. The suppression of magnetic order is further enhanced with increased Ni doping, leading to a reentrant nonmagnetic state below TC for x=1.3%. The reentrant behavior indicates an interplay between antiferromagnetism and superconductivity involving competition for the same electrons. These observations are consistent with the sign-changing s± superconducting state, which is expected to appear on the verge of microscopic coexistence and phase separation with magnetism. We also present a universal linear relationship between the local ordered moment size and the antiferromagnetic ordering temperature TN across a variety of iron-based superconductors. We argue that this linear relationship is consistent with an itinerant-electron approach, in which Fermi surface nesting drives antiferromagnetic ordering. In studies of superconducting properties, we find that the T=0 limit of superfluid density follows the linear trend observed in underdoped cuprates when plotted against TC. This paper also includes a detailed theoretical prediction of the muon stopping sites and provides comparisons with experimental results.